6. (a) Write $ \sqrt{80} $ in the form $ c \sqrt{5} $, where $ c $ is a positive constant. (b) A rectangle R has a length of $ (1 + \sqrt{5}) $ cm and an area of $ \sqrt{80} $ cm². (b) Calculate the width of R in cm. Express your answer in the form $ p + q\sqrt{5} $, where $ p $ and $ q $ are integers to be found.

A-Level Edexcel Maths Pure Exponential & Logarithms: 6. (a) Write $ \sqrt{80} $ in

6. (a) Write $ \sqrt{80} $ in the form $ c \sqrt{5} $, where $ c $ is a positive constant - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 1

Question 8

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6. (a) Write $ \sqrt{80} $ in the form $ c \sqrt{5} $, where $ c $ is a positive constant. (b) A rectangle R has a length of $ (1 + \sqrt{5}) $ cm and an ar... show full transcript

Worked Solution & Example Answer:6. (a) Write $ \sqrt{80} $ in the form $ c \sqrt{5} $, where $ c $ is a positive constant - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 1

Step 1

Write $ \sqrt{80} $ in the form $ c\sqrt{5} $

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Answer

To express ( \sqrt{80} ) in the form ( c\sqrt{5} ), we first factor ( 80 ) as follows:

[ 80 = 16 \times 5 ]

Then, taking the square root:

[ \sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \cdot \sqrt{5} = 4\sqrt{5} ]

Thus, ( c = 4 ).

Step 2

Calculate the width of R in cm. Express your answer in the form $ p + q\sqrt{5} $

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Answer

Given that the length of rectangle R is ( (1 + \sqrt{5}) ) cm and the area is ( \sqrt{80} ) cm², we need to find the width ( w ).

The area can be expressed as:

[ \text{Area} = \text{Length} \times \text{Width} \implies \sqrt{80} = (1 + \sqrt{5}) w ]

To find ( w ), rewrite the equation:

[ w = \frac{\sqrt{80}}{1 + \sqrt{5}} ]

Next, we rationalize the denominator:

[ w = \frac{\sqrt{80}}{1 + \sqrt{5}} \cdot \frac{1 - \sqrt{5}}{1 - \sqrt{5}} = \frac{\sqrt{80}(1 - \sqrt{5})}{1 - 5} = \frac{\sqrt{80}(1 - \sqrt{5})}{-4} ]

We know that ( \sqrt{80} = 4\sqrt{5} ), so substituting this value gives:

[ w = \frac{4\sqrt{5}(1 - \sqrt{5})}{-4} = -\sqrt{5}(1 - \sqrt{5}) ]

This simplifies to:

[ w = -\sqrt{5} + 5 = 5 - \sqrt{5} ]

Thus, in the form ( p + q\sqrt{5} ), we have ( p = 5 ) and ( q = -1 ).

6. (a) Write \( \sqrt{80} \) in the form \( c \sqrt{5} \), where \( c \) is a positive constant - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 1

Worked Solution & Example Answer:6. (a) Write \( \sqrt{80} \) in the form \( c \sqrt{5} \), where \( c \) is a positive constant - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 1

Write \( \sqrt{80} \) in the form \( c\sqrt{5} \)

Calculate the width of R in cm. Express your answer in the form \( p + q\sqrt{5} \)

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